Let `n` be a positive integer, `q=2^n`, and let `\mathbb{F}_q` be the finite field with `q` elements. For each positive integer `m`, let `D_m(X)` be the Dickson polynomial of the first kind of degree `m` with parameter `1`. Assume that `m>1` is a divisor of `q+1`. We study the existence of `\alpha\in\mathbb{F}_q^\ast` such that `D_m(\alpha)=D_m(\alpha^{-1})=0`. We also explore the connections of this question to an open question by Wiedemann and a game called ''Button Madness''.
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